Jessica Taylor. CS undergrad and Master’s at Stanford; former research fellow at MIRI.
I work on decision theory, social epistemology, strategy, naturalized agency, mathematical foundations, decentralized networking systems and applications, theory of mind, and functional programming languages.
Blog: unstableontology.com
Twitter: https://twitter.com/jessi_cata
Ah, I was confused by statements such as the following:
Now, imagine if all of those developments were instead compressed into the decade after 1925.
Now imagine all the scientific, intellectual and technological developments that you would expect to see by the year 2125, if technological progress continued over the next century at roughly the same rate that it did over the last century. And then imagine all of those developments occurring over the course of just ten years.
Have edited it to one about training scale-ups. “The biggest training runs can continue to scale for roughly another 10,000x increase before running into power and other limits (likely well within a decade from now)...”
Re pagerank: This looks a lot like eigenvalues / eigenvectors, which show up a bunch in physics. (Eigenvector with high eigenvalue is like a “self-ratifying / stable generalized state”.)
Quantum mechanics involves topology of course. In operator algebra theory (Gelfand duality). And in TQFT. However the article seems to be making a leap relating quantum topology to minds and phenomenal binding. This is skipping many levels of abstraction!
I will now summarize a quantum topology approach, relevant to observers, which is not skipping nearly as many abstraction levels. We start by a quantum operator space as a C* algebra. This algebra is in general non-commutative. However, it has commutative sub-algebras. (Sub-algebra here is similar to ‘sub-group’, ‘sub-monoid’, ‘category theoretic sub-object’; has a precise characterization). These form a meet-semilattice (indicating: ‘intersection’ of commutative sub-algebras is commutative; ‘union’ is not in general commutative). The meet-semilattice structure reflects complementarity, Heisenberg uncertainty, Kochen-Specker, and so on. In PVM/POVM terms, different Hermitian operators commute when they each have a “diagonalization” (or infinite-dimensional equivalent) in a compatible basis; this is not always the case.
So we have a meet-semilattice of commutative C* sub-algebras. Then each, by Gelfand duality, is iso to a C* algebra of continuous functions for a compact Hausdorff space . Accordingly, there is a category-theoretic contravariant isomorphism between the category of commutative C* algebras and the category of compact Hausdorff spaces and continuous maps between them.
This is of course highly topological. A quantum operator algebra implies multiple classical contexts, and in general there’s no classical context containing all information from all of them. The contexts are at varying levels of fine-ness and coarse-ness. Some coarsening is necessary to get commutativity (and classicality, under the ‘commutative C* sub-algebras as classical’ interpretation.)
The sub-algebraic picture suggests that ‘high-level computations’ can be really instantiated as sub-algebras. Where sub-algebras also relate to topology through locale theory; categorical sub-objects in a category such as the category of compact regular frames. (Compare: if a group has a sub-group isomorphic to under addition, then the group operation implements integer addition; this is a hard mathematical constraint, not merely an interpretation.)
There is some ‘reality to wholes’ here, through sub-algebras (corresponding with quotient spaces in topology through locale-theoretic duality, Isbell). There is some ‘objectivity’ here (or perhaps ‘pre-conditions for objectivity’), in that which coarsenings form valid commutative algebras depends on the physical system in question.
The conflation to avoid making here is that a classical context (given by a compact Hausdorff space corresponding to a commutative C* sub-algebra) is ‘a mind’ or ‘a person’ or that sort of thing. It can be much more detailed than that. It is more like a virtual world simulation that doesn’t exactly have a reductionist lowest level to it; some details are simply coarsened. A given classical context can contain multiple minds (as classical reductionists expected prior to quantum mechanics). If anything, the classical ‘mind vs matter’ distinction is in a frame that makes classical assumptions; a classical context is more like a pre-requisite for ‘mind vs matter’ to be a sensible distinction. (Materialism != physicalism)
This is more my own philosophical spin than something directly implied by quantum topology, but: The ‘phenomena’ here are more like the phenomena of Kant than the phenomena of Chalmers: spatially three-dimensional, multi-personal. See also Wilfrid Sellars on phenomena; his “Empiricism and the Philosophy of Mind” is of course important background, but his “Phenomenalism” addresses multi-personal phenomenal contexts more directly.
(See Bohrification and a review for more technical details on this overall picture.)
To disambiguate terminology: By “dual space” I mean the standard meaning in QM; we go from a complex vector space V to a space of linear maps , with the bra/ket duality as a special case. Perhaps I should avoid using “dual” but this explains my earlier usage.
By “complex conjugate space” (sometimes abbreviated as “conjugate space”) I mean specifically the construction, a completely formal mathematical operation. (To avoid confusion I could say “complex conjugate space”)
Since “complex conjugate space” is an entirely formal construction, it doesn’t necessarily have a physical meaning. Or it could have multiple physical meanings as different isos with V.
OP doesn’t try to go lower-level than Hilbert space; what follows is my attempt to engage with the level you’re talking about:
A “duality” like thing is “polarity of representation of ” as implied by the category . Where to simplify in OP, I’m always using the polarity between V and in the representation, but this is not strictly necessary.
The correspondence with the SO(n) model might be: When you are representing an element of as a complex vector space ( means “delooping groupoid”), you are picking out a sub-group of SO(n) iso to . To get to a Hilbert space you clearly have to at least pick out a subgroup.
Another idea is to find a functor . This does not necessarily get you an O(2) subgroup, because J and in need not map to the same group element of O(n). (Hence spinors and so on)
If you have a representation and a groupoid functor you can trivially compose to get a polar representation . That gives you, as the positive component, something like a Hilbert space, and as the negative component, something formally iso to its complex conjugate space (though with a better physical interpretation).
At this point you can use the formal isos: (Riesz representation), and the iso from to the polar negative of your Hilbert-like space (where polar negation is from the representation). This gives you nice physical interpretations of bra/kets, density matrices, observables, and so on, through formal isos.
(I am not sure how much I’m understanding or how much is connecting; feel free to ignore irrelevant detail)
In terms of goals of OP, I was taking standard QM notation at face value, and trying to peel back one layer of the onion: why antilinear-linear inner product, POVM observables as Hermitians implying quadratic forms, quadratic density matrices, etc? And I think real structure + conjugate spaces basically explains these dualities in the Hilbert space standard formulation, although that doesn’t mean this is the deepest symmetry layer (it’s not).
field strength tensor
This is not an area I understand well, but, either the field values are expressed as a real number or complex number. If real: OP framework doesn’t apply, or applies at a lower level; perhaps applies indirectly through CPT. If complex: Then I’m guessing similar principles will apply, where for example you can “make a complex number real” as , yielding an inner product that looks like the Hilbert space inner product. The thing to look for would be, are complex conjugates showing up all over the place in the standard math? If so, it may perhaps be modeled with real structure & complex conjugate spaces.
we should not expect to be able to find dual spaces by looking at mirror images
For Hilbert space I’m applying Riesz representation for that; standard machinery for bra-ket. Not sure how far the principle generalizes. (Much of OP could work without Riesz representation, e.g. we can still define just fine, it’s just that might not be an invertible map.)
We can identify
This is looking a bit like the ‘swap’ real structure on . The main difference is that it’s dealing with real rather than complex numbers. With the swap real structure, the ‘imaginary component’ would be written as . This of course looks similar to .
Here’s one place where the analogy with symmetric group permutations breaks down. Say our Hilbert space is . Now our inner product (written to be bilinear) is . We can see this as quadri-linear in . Notice with ‘n-linearity’ we always have n even, and the elements come in pairs. We pair with and with .
What the standard Hilbert space framework is going to do is, by bundling , make the and bilinearity explicit, with the and bilinearity implicit (through the inner product and bra/kets and so on). This is why we end up with quadratics all over the formalism: Born rule, inner product, density matrices, POVMs.
So for your analogy to hold, it’s critical that be even. And what the ‘swap’ real structure on is going to do is swap elements with their conjugated elements ( with , and with ). This is going to be a specific involutive permutation.
So I guess if you wanted to get into deeper foundations, then a question to ask would be “why do the tensor products that POVM observables are linear in factor into components that come in pairs with their complex conjugate spaces (or something iso to their complex conjugate spaces)?”. Which, again, OP isn’t very much about.
The bilinear form is from the inner product. The inner product is generally defined as ‘antilinear in first argument, linear in second’. If replacing the first argument with complex conjugate space, it is now ‘linear in first argument, linear in second argument’, i.e. bilinear. This is of course for a single Hilbert space (and its complex conjugate). When having multiple Hilbert spaces tensored together, the ‘tensor product of Hilbert spaces’ yields an inner product for that tensored space. That is implicitly multilinear, although decomposes as usual for tensor product, into multiple bilinearities.
Here’s an evalution claiming it isn’t even true at Anthropic in terms of lines of code (I haven’t checked independently)
The kind of thing I don’t want to count is that, for example, it would be easy to write a script that generates many many lines of code. It’s just not interesting to count “how are most lines of code written” globally. “Code actually merged at particular big tech companies” is much closer and I expect that was less than 90 percent at the time.
Ah sorry. Edited.
With 10K lines it’s not impressive to write 10K lines. I’m thinking of writing software that would be 10K lines if done by humans.
“bug-free” & “reliably” is a high bar!
I think the idea of paying attention to “human values” and “alignment” was based on a philosophical assumption that moral realism would fail, that value is relative / two-place predicate. I am thinking perhaps a more disjunctive approach would be to imagine an AI that in the case of moral realism does the normative thing and in the case of moral non-realism is aligned with humans / humanity / something Earth-local. This assumes there is a fact of the matter about whether moral realism is true. Also, not everyone will like it even if moral realism is true (though not everyone liking something is a usual state of affairs). Even framing it disjunctively like this is perhaps philosophically loaded / hard to define without better concepts. See also Moral Reality Check short story about what if AIs believe moral realism.
Really appreciate the exchange, helped me clarify my thoughts.
As a brief note for potential future exploration (no need to reply if you don’t want to):
A crux seems to be that with chroma world, I think experience is of difference between chroma, assuming chromatic realism (real quiddites). Whereas the Russelian monist wants the experience to be of identity of internal chroma. This is perhaps a disagreement on the phenomenology / epistemology relationship.
I think the analogy with Newtonian mechanics (if absolute positions are real, then experience is of relative positions, even if located at absolute positions) is pretty strong; chroma world “just” has different physical symmetries.
On my end I think I could continue this line of exploration by studying Deleuze (difference as more ontologically/metaphysically primary than identity), Dennett (more clarity on whether I agree/disagree with his basic picture), and Russell as a primary source.
Yeah I think it might be more productive, if I wanted to make progress on this, to look at the math rather than the metaphysical back-and-forth.
I feel like the single particle universe clearly instantiates the logical system: ∃x (particle(x)).
Nope, because they didn’t write down the system. (This might be quibbling. I mean it doesn’t write down that formal system anywhere.)
Let me move to a different example, why do you think our universe instantiates PA
Because mathematicians have written down the rules of PA. And they have programmed computers to use PA.
Do you think that our universe then either instantiates Con(ZFC) or doesn’t?
Our universe instantiates formal system such as ZFC + Con(ZFC), because mathematicians can write them down. (It also instantiates inconsistent formal systems such as “ZFC + axiom of choice is false”.)
Backtracking a bit. I reject axiom 6 (which is not to say I actively disbelieve it, but that I don’t actively believe it) and I’m giving an idea of why. If you could logically prove axiom 6, you wouldn’t need it to be an axiom. So you’re going to have some trouble convincing me of it on logical grounds. Rather it’s more metaphysical and so on.
Let us say that PA consists of a formal system with exactly the standard axioms of PA. There is some quibbling about which “paper copies” work or which “translations” work. But PA only exists in a universe if it is instantiated as physical information. Instantiating large numbers does not instantiate PA.
Okay, noting that you made a stronger claim, that there is a universe where there is no sound logical system (i.e. P and not P are true in some possible universe), but I’m happy to move to this, sure.
Not what I meant. Two interpretations (a) in the single-particle universe, no logical system, sound or unsound, is instantiated, because no non-trivial physical info is instantiated. (b) Perhaps we could say that PA’s prefect-generating property is stronger than soundness. It is soundness plus “having provably distinct terms that number in the countable infinity”.
Imagine a universe that consists of a single particle. The law is that the particle stays in the same state always. Actually, the particle only has 1 possible state in this universe. So it’s just always in the same state as a matter of math not just physics.
There is no way for PA to exist in this universe.
“PA is possible” sure ok.
“PA has a perfect generating property” let is suppose this. for example, maybe there is some list of perfect properties of PA, and they all follow from some property of it.
“It is possible for that perfect-generating property to be instantiated (it is instantiated in Peano Arithmetic)” sure, ok
“Therefore it is possible for something to have the property of perfect-essential necessity.” absolutely wild claim.
“We say that something has the property of “perfect-essential necessity” if (and only if) whenever there is a perfect-generating property for that thing, it implies that there necessarily must exist something (i.e. in every world) which instantiates that perfect-generating property.”
Seems like a wild claim. For PA to have this property would mean that its perfect-generating property (perhaps logical consistency, or soundness?) has a corresponding object having that property (e.g. soundness) in every possible world. But that’s a wild claim. Maybe there’s a possible universe with so little information that no sound logical system exists there.
Idk. But suppose there is an essence that generates all perfect properties of PA. How would “perfect-essential necessity is possible” follow? (And if it does follow, why do you need axiom 6?)
Yes this is a good approximation.
It’s complicated by i appearing in the Schrodinger equation. Which is essentially a convention. The thing close to ‘replace i with -i in states’ that would produce a real symmetry is CPT.
One thing that quotients out some of this info is the density matrix |ψ⟩⟨ψ|, corresponding to ¯¯¯v⊗v in the post. If H is a Hilbert space, operators H→H have a real structure given by the Hermitian adjoint; and density matrices by definition satisfy conditions including being self-adjoint (i.e. Hermitian, i.e. ρ†=ρ). This means density matrices ‘are real’ in the relevant sense, analogous to real numbers. (Operators representing POVM observables are also self-adjoint/Hermitian.) On the other hand, the derivative of unitary time evolution U’(0) is ‘entirely imaginary’, which is related to the ‘i’ in the Schrodinger equation being conventional.